The function estimates, predict and forecast time series data with models,
and also make subset forecasts within the length of the entire trend of the
data. However, the forecast is constrained to lower and upper 80% and 95% forecasts of the
of the data for integer series in line with Hyndman & Athanasopoulos (2021).
The recognized models are lm, smooth spline, polynomial splines with or
without knots, quadratic polynomial, and ARIMA. The robust output include
the models' estimates, time-varying forecasts and plots based on themes
from ggplot. The main attraction of this function is the use of the newly
introduced equal number of trend to forecast from the model.
The function takes daily, monthly and yearly data sets for now.
Usage
DynamicForecast(Data, date, series, dyrima, Trend, Type, MaximumDate, x = 0,
x100 = 0, BREAKS = 0, ORIGIN = NULL, origin = "1970-01-01", Length = 0, ...)Arguments
- date
A vector containing the dates for which the data is collected. Must be the same length with
series. The date must be in 'YYYY-MM-DD'. If the data is monthly series, the recognized date format is the last day of the month of the dataset e.g. 2021-02-28. If the data is a yearly series, the recognized date format is the last day of the year of the data set e.g. 2020-12-31. There is no format for Quarterly data for now.- x
![[Deprecated]](figures/lifecycle-deprecated.svg)
- series
A vector containing observations for estimation and forecasting. Must be the same length with
date.- dyrima
ARIMA object of the
seriesobtained fromauto.rimain forecast package.- x100
vector of optional dataset that is to be added to the model for forecasting. The modeling and forecasting is still done if not provided. Must be the same length with
series.- BREAKS
A vector of numbers indicating points of breaks for estimation of the spline models.
- MaximumDate
- . The date indicating
the maximum date (last date) in the data frame, meaning that forecasting
starts the next date following it. The date must be a recognized date
format. Note that for forecasting, the date origin is set to 1970-01-01.
- Trend
The type of trend. There are three options Day, Month and Year.
- Type
The type of response variable. There are two options Continuous and Integer. For integer variable, the forecasts are constrained between the minimum and maximum value of the response variable.
- Length
The length for which the forecast would be made. If not given, would default to the length of the dataset i.e. sample size.
- origin
default date origin which is 1970-01-01 used to position the date of data so that the forecasts are in tandem with the period of the observations.
- ORIGIN
date origin of the dataset and if different from origin must be in the format
"YYYY-MM-DD". This is used to position the date of the data to properlydatethe forecasts.- Data
- . Now broken into three
vectors
date,seriesandx100. - ...
Additional arguments that may be passed to the function.
Value
A list with the following components:
Spline without knotsThe estimated spline model without the breaks (knots).
Spline with knotsThe estimated spline model with the breaks (knots).
Smooth SplineThe smooth spline estimates.
ARIMAEstimated Auto Regressive Integrated Moving Average model.
QuadraticThe estimated quadratic polynomial model.
Ensembled with equal weightEstimated Ensemble model with equal weight given to each of the models. To get this, the fitted values of each of the models is divided by the number of models and summed together.
Ensembled based on weightEstimated Ensemble model based on weight of each model. To do this, the fitted values of each model served as independent variable and regressed against the trend with interaction among the variables.
Ensembled based on summed weightEstimated Ensemble model based on summed weight of each model. To do this, the fitted values of each model served as independent variable and is regressed against the trend.
Ensembled based on weight of fitEstimated Ensemble model. The fit of each model is measured by the rmse.
Unconstrained ForecastThe forecast if the response variable is continuous. The number of forecasts is equivalent to the length of the dataset (equal days forecast).
Constrained ForecastThe forecast if the response variable is integer. The number of forecasts is equivalent to the length of the dataset (equal days forecast).
RMSERoot Mean Square Error (rmse) for each forecast.
Unconstrained forecast PlotThe combined plots of the unconstrained forecasts using ggplot.
Constrained forecast PlotThe combined plots of the constrained forecasts using ggplot.
DateThis is the date range for the forecast.
Fitted plotThis is the plot of the fitted models.
Estimated coefficientsThis is the estimated coefficients of the various models in the forecast.
Examples
library(readr)
library(forecast)
COVID19$Date <- zoo::as.Date(COVID19$Date, format = '%m/%d/%Y')
#The date is formatted to R format
LEN <- length(COVID19$Case)
Dss <- seq(COVID19$Date[1], by = "day", length.out = LEN)
#data length for forecast
ORIGIN = "2020-02-29"
lastdayfo21 <- Dss[length(Dss)] # The maximum length # uncomment to run
Data <- COVID19[COVID19$Date <= lastdayfo21 - 28, ]
# desired length of forecast
BREAKS <- c(70, 131, 173, 228, 274) # The default breaks for the data
dyrima <- auto.arima(Data$Case)
DynamicForecast(date = Data$Date, series = Data$Case, dyrima = dyrima,
BREAKS = BREAKS, Trend = "Day", Length = 0, Type = "Integer", x100 = 0)
#> Warning: Coercing LHS to a list
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> $`Spline without knots`
#>
#> Call:
#> stats::lm(formula = series ~ splines::bs(Series, knots = NULL))
#>
#> Coefficients:
#> (Intercept) splines::bs(Series, knots = NULL)1
#> -396.5 2212.6
#> splines::bs(Series, knots = NULL)2 splines::bs(Series, knots = NULL)3
#> -912.1 1621.2
#>
#>
#> $`Spline with knots`
#>
#> Call:
#> stats::lm(formula = series ~ splines::bs(Series, knots = BREAKS))
#>
#> Coefficients:
#> (Intercept) splines::bs(Series, knots = BREAKS)1
#> 190.1 -634.7
#> splines::bs(Series, knots = BREAKS)2 splines::bs(Series, knots = BREAKS)3
#> 1742.0 -1477.3
#> splines::bs(Series, knots = BREAKS)4
#> 1199.9
#>
#>
#> $`Smooth Spline`
#> Call:
#> stats::smooth.spline(x = Series, y = series)
#>
#> Smoothing Parameter spar= 0.4790742 lambda= 5.206569e-06 (12 iterations)
#> Equivalent Degrees of Freedom (Df): 32.10921
#> Penalized Criterion (RSS): 7579996
#> GCV: 29266
#>
#> $ARIMA
#> Time Series:
#> Start = 1
#> End = 320
#> Frequency = 1
#> [1] 0.99900000 0.84528948 0.23242234 0.19426066 0.23530389
#> [6] 0.18708366 0.20645071 0.09392626 -0.05029039 -0.12751880
#> [11] -0.09508467 0.46504846 0.12239000 0.14548861 0.20729516
#> [16] 0.08546458 0.10439869 0.05184082 -0.02306526 -0.05671365
#> [21] 2.82802596 2.35418907 5.57152616 3.52213383 7.02699363
#> [26] 8.18974689 7.33454460 13.04482986 18.93331641 13.42789704
#> [31] 16.73706721 11.56629015 27.70845053 17.36076948 22.02026457
#> [36] 15.13295123 15.08353256 11.95120672 14.01732590 29.72495102
#> [41] 19.16495321 15.76480194 19.10329764 13.67793324 18.35549181
#> [46] 14.57291651 16.88749994 18.96793027 19.23755469 23.57464546
#> [51] 35.49894393 59.90850008 44.76916782 84.95613619 80.04915644
#> [56] 86.76356517 101.98862342 90.91460794 99.02262416 88.23105898
#> [61] 145.59796785 150.60421503 154.52368562 187.64195367 191.43146550
#> [66] 190.09834215 243.00601377 201.72881635 208.39385336 288.70681193
#> [71] 282.49818424 230.78804155 259.64196433 273.87659408 238.73752450
#> [76] 256.08056296 228.28610073 226.67871401 159.36620385 232.94066634
#> [81] 213.03084814 237.13647050 300.86417459 315.43530480 265.57250383
#> [86] 258.29599042 275.34920247 240.01517161 281.66646401 347.85779215
#> [91] 245.48719089 331.73071905 416.77976221 322.31902239 389.26287681
#> [96] 332.11617117 372.70080701 394.25839547 354.21033868 363.78953368
#> [101] 274.24946690 289.41558654 480.02191952 396.30190817 551.69228445
#> [106] 570.97712445 497.71786106 497.40000084 575.40905043 539.34952603
#> [111] 566.93276948 631.60522920 573.29293903 593.39316916 532.77037276
#> [116] 666.17839729 592.38582758 646.66889702 605.16286479 579.12335338
#> [121] 640.84172837 529.23394052 608.99965448 643.49167033 732.00676152
#> [126] 654.41231894 522.99023142 570.87053287 535.40644280 576.17524301
#> [131] 578.62219750 531.20590822 519.26387006 524.08878948 562.19156012
#> [136] 538.12598299 570.61603561 533.91485307 620.63549290 613.17827696
#> [141] 596.69908573 616.18007719 551.48003872 562.38882391 591.65510906
#> [146] 583.20969192 612.75219503 591.09678438 494.90732111 536.71880960
#> [151] 584.38416520 583.19361950 503.82932047 528.73980185 506.11203767
#> [156] 448.04403350 403.71674706 359.04013504 331.27550007 382.91747471
#> [161] 336.00606824 378.33311297 401.86602629 407.37011537 370.39093072
#> [166] 468.16921168 417.38872010 388.14616651 367.81361127 335.42499378
#> [171] 325.15498520 382.09614995 385.37022332 468.01781911 431.46645076
#> [176] 372.36965178 525.45899741 415.79972151 414.09932639 383.96878188
#> [181] 295.73338550 301.13900684 205.51646870 225.91423502 178.55935487
#> [186] 169.58991049 229.57139296 208.77196905 160.02360708 165.03499862
#> [191] 157.20925222 128.14111654 162.80347023 227.95674232 167.94033105
#> [196] 177.67765611 184.48135296 170.87919338 152.33212245 172.71690972
#> [201] 135.56846047 122.04732303 111.30930055 142.30424489 145.73504847
#> [206] 121.73954336 188.51488855 187.46584108 151.39917761 152.80497228
#> [211] 172.55221920 129.17259956 129.23021921 145.14873964 167.85613232
#> [216] 183.92759868 162.22400941 144.61878956 154.72792301 106.85464470
#> [221] 135.04631390 134.57630232 133.47075972 107.50797845 120.07927499
#> [226] 109.62956728 140.46851998 159.86280830 189.49693411 174.33170093
#> [231] 225.36258131 200.35053496 158.62453022 184.17536367 169.39294337
#> [236] 134.09652740 96.92174162 110.59602125 92.57339537 49.76618208
#> [241] 71.70005198 108.33012862 113.00172536 134.78096219 131.13037961
#> [246] 129.98770066 134.02812607 124.46015053 121.21974798 220.03979828
#> [251] 165.28185388 162.68323775 188.32528871 99.69249400 229.18205838
#> [256] 158.67648312 170.14396194 195.24157908 173.32460200 153.43000269
#> [261] 131.91940098 155.57072589 162.68808740 163.57955745 200.97807628
#> [266] 150.09933218 140.54555316 199.91071148 165.61015497 130.35327723
#> [271] 182.88419865 178.77739929 153.16700120 192.38042601 128.38426664
#> [276] 117.62437134 165.75952493 227.75615894 157.62623637 246.33450167
#> [281] 242.45601936 235.86261688 294.21846852 353.84868769 457.79175055
#> [286] 442.39559357 768.28265853 1588.96845988 745.26562871 753.13203313
#> [291] 814.94237712 939.24280630 1092.11260196 1011.68845314 649.60974152
#> [296] 560.70516352 483.86504471 611.77867970 1104.43324249 1152.92374466
#> [301] 1008.14941332 675.06258902 634.58241723 749.67656294 705.70389089
#> [306] 975.96296794 1042.91314029 869.68620342 811.23913958 561.45880246
#> [311] 798.60501240 1107.98586199 1298.37506527 1467.19527431 1368.61156068
#> [316] 1327.56147075 1426.11011520 1289.77484996 1483.67880299 1503.24004691
#>
#> $Quadratic
#>
#> Call:
#> stats::lm(formula = series ~ Series + I(Series^2))
#>
#> Coefficients:
#> (Intercept) Series I(Series^2)
#> 1.479e+02 1.504e-01 4.727e-03
#>
#>
#> $`Ensembled with equal weight`
#> Time Series:
#> Start = 1
#> End = 320
#> Frequency = 1
#> 1 2 3 4 5 6
#> -11.3989579 -12.6647237 -13.7978782 -14.5958942 -15.1621183 -15.5344936
#> 7 8 9 10 11 12
#> -15.6859007 -15.6603546 -15.4414091 -15.0127902 -14.3690701 -13.4283285
#> 13 14 15 16 17 18
#> -12.4779949 -11.2655119 -9.8567243 -8.2963564 -6.5205721 -4.5760214
#> 19 20 21 22 23 24
#> -2.4578630 -0.1592426 2.8858808 5.4118496 8.8181602 11.3020224
#> 25 26 27 28 29 30
#> 15.0162336 18.3688704 21.4070989 25.8286905 30.3398404 32.6200564
#> 31 32 33 34 35 36
#> 36.7075364 39.1411015 45.8819752 47.3734777 51.9193157 54.2147032
#> 37 38 39 40 41 42
#> 57.9430188 61.1258291 65.4222870 72.5231764 74.4549730 77.9359912
#> 43 44 45 46 47 48
#> 82.9209414 86.3431780 91.9895347 96.1557826 101.7569108 107.5107792
#> 49 50 51 52 53 54
#> 113.0813273 119.6221586 127.8098860 138.5954630 141.5478807 155.6359053
#> 55 56 57 58 59 60
#> 160.7736920 168.3079848 177.6421981 181.8454208 190.0203033 194.4585354
#> 61 62 63 64 65 66
#> 212.4551362 219.8033232 226.7094642 239.1959015 245.5268953 250.5316152
#> 67 68 69 70 71 72
#> 266.0752364 262.4494099 267.9935194 287.7469601 289.6251508 281.9856768
#> 73 74 75 76 77 78
#> 290.2462877 295.5524678 291.0455804 297.1665539 294.4413748 297.1054310
#> 79 80 81 82 83 84
#> 286.7308970 304.5915760 303.7876813 311.7983909 327.7242670 333.7965786
#> 85 86 87 88 89 90
#> 326.9779189 328.6981626 335.3229593 331.4904017 343.0476208 359.4648882
#> 91 92 93 94 95 96
#> 342.0279863 362.0650139 381.5554949 364.9710558 380.6776181 371.7471481
#> 97 98 99 100 101 102
#> 382.6668281 390.1957336 385.8931274 391.9374873 378.4756379 386.1441211
#> 103 104 105 106 107 108
#> 428.8624937 416.4255203 451.3314367 458.5992016 447.0660584 449.9165353
#> 109 110 111 112 113 114
#> 468.1650422 463.2322606 470.5817281 484.9099651 474.2308995 478.8656741
#> 115 116 117 118 119 120
#> 467.0807369 493.9333823 479.2466377 489.9962028 481.2923679 475.2986347
#> 121 122 123 124 125 126
#> 486.5022082 462.7443489 477.0184207 482.0298565 497.6682443 479.9557807
#> 127 128 129 130 131 132
#> 451.4715028 458.9825616 450.0863285 456.7673196 456.1692305 446.0056216
#> 133 134 135 136 137 138
#> 443.2346207 443.9725950 451.3975648 446.3681942 452.5951981 444.8857169
#> 139 140 141 142 143 144
#> 461.6812508 459.3713469 454.9144130 457.3051086 442.5311681 442.5620655
#> 145 146 147 148 149 150
#> 445.9411443 441.4436987 444.1951487 436.3292674 413.1461276 417.1402016
#> 151 152 153 154 155 156
#> 421.9502247 416.7225664 395.7091867 395.6241545 386.3564343 370.5389326
#> 157 158 159 160 161 162
#> 358.0680883 346.1478755 338.2237178 346.6724254 335.7510644 342.8670651
#> 163 164 165 166 167 168
#> 346.2884162 346.1444494 337.5459862 355.9501219 344.7315008 337.9545933
#> 169 170 171 172 173 174
#> 333.1028930 325.8480838 322.8539495 332.9606305 331.7852121 345.7213333
#> 175 176 177 178 179 180
#> 334.9020759 318.7539794 344.3812546 316.9698230 310.8939758 299.1236767
#> 181 182 183 184 185 186
#> 275.9465442 271.8334687 247.9471206 247.7729617 234.5492234 229.4756069
#> 187 188 189 190 191 192
#> 238.6377847 232.0639227 220.2954367 219.6349011 216.6670295 209.6006773
#> 193 194 195 196 197 198
#> 215.3292425 227.1392980 213.8421269 214.3952378 214.3516662 210.3368961
#> 199 200 201 202 203 204
#> 205.5414973 208.7530086 200.6690468 197.5008558 195.0127235 200.9162459
#> 205 206 207 208 209 210
#> 201.2887414 196.1549746 209.1666896 208.6151003 201.0636235 201.0102851
#> 211 212 213 214 215 216
#> 204.6275696 195.6151968 195.2780459 198.0967743 202.2683710 205.1231378
#> 217 218 219 220 221 222
#> 200.4622510 196.7443591 198.7903630 189.5387261 195.7971885 196.5988051
#> 223 224 225 226 227 228
#> 197.5191522 193.6520961 197.6060738 196.9877004 204.5263729 209.5303485
#> 229 230 231 232 233 234
#> 216.2186355 213.5742456 223.8143100 218.5397452 209.7878218 214.5588643
#> 235 236 237 238 239 240
#> 211.5107976 204.6985303 197.9180228 201.7685570 199.7345704 193.1771179
#> 241 242 243 244 245 246
#> 199.9799246 210.0985073 214.1100244 221.7216966 224.3364645 227.5180488
#> 247 248 249 250 251 252
#> 231.7941019 233.3902512 236.2428857 259.4390919 251.8058321 254.5017863
#> 253 254 255 256 257 258
#> 262.7685787 248.1280430 277.0828764 266.0324152 271.3929670 279.5179499
#> 259 260 261 262 263 264
#> 278.3002510 277.5657327 276.5917219 284.7354412 289.6547368 293.3897964
#> 265 266 267 268 269 270
#> 304.4611690 297.9074838 299.7308013 315.5530616 312.9605542 310.5986742
#> 271 272 273 274 275 276
#> 326.3198364 331.3614663 332.9486738 348.5641676 344.8047974 353.1084915
#> 277 278 279 280 281 282
#> 374.7420722 400.7940902 402.0029979 436.4580027 453.7359608 471.5354186
#> 283 284 285 286 287 288
#> 503.0789765 535.1551709 575.2302046 589.2659210 668.4291491 843.8656718
#> 289 290 291 292 293 294
#> 684.5307801 694.3226320 714.0780302 745.7818857 782.9192597 773.3607881
#> 295 296 297 298 299 300
#> 707.6711685 697.0152828 689.2196223 722.8327933 829.8662208 848.5658690
#> 301 302 303 304 305 306
#> 829.1417881 772.6884855 775.7891676 811.5105791 817.2812117 887.6515257
#> 307 308 309 310 311 312
#> 918.8876439 903.3891201 911.9163096 883.0144827 951.9989884 1035.1767707
#> 313 314 315 316 317 318
#> 1093.4433832 1145.6228474 1142.6823890 1149.9424089 1184.1562352 1170.7973009
#> 319 320
#> 1223.2676078 1241.0281464
#>
#> $`Ensembled based on weight`
#>
#> Call:
#> stats::lm(formula = Series ~ Without.knots * With.knots * Smooth *
#> Quadratic * ARIMA)
#>
#> Coefficients:
#> (Intercept)
#> -1.041e+02
#> Without.knots
#> 2.955e-01
#> With.knots
#> -1.379e-01
#> Smooth
#> -7.582e-02
#> Quadratic
#> 8.501e-01
#> ARIMA
#> -3.154e-02
#> Without.knots:With.knots
#> -1.730e-05
#> Without.knots:Smooth
#> 3.781e-04
#> With.knots:Smooth
#> -2.171e-04
#> Without.knots:Quadratic
#> -1.569e-03
#> With.knots:Quadratic
#> 1.052e-03
#> Smooth:Quadratic
#> 2.033e-04
#> Without.knots:ARIMA
#> 1.379e-04
#> With.knots:ARIMA
#> -6.588e-05
#> Smooth:ARIMA
#> 4.391e-04
#> Quadratic:ARIMA
#> 8.397e-05
#> Without.knots:With.knots:Smooth
#> 5.938e-09
#> Without.knots:With.knots:Quadratic
#> 1.307e-08
#> Without.knots:Smooth:Quadratic
#> -1.249e-06
#> With.knots:Smooth:Quadratic
#> 8.810e-07
#> Without.knots:With.knots:ARIMA
#> -2.892e-09
#> Without.knots:Smooth:ARIMA
#> -2.085e-06
#> With.knots:Smooth:ARIMA
#> 1.123e-06
#> Without.knots:Quadratic:ARIMA
#> -4.872e-07
#> With.knots:Quadratic:ARIMA
#> 3.261e-07
#> Smooth:Quadratic:ARIMA
#> -1.140e-06
#> Without.knots:With.knots:Smooth:Quadratic
#> -2.182e-11
#> Without.knots:With.knots:Smooth:ARIMA
#> 7.294e-11
#> Without.knots:With.knots:Quadratic:ARIMA
#> 8.872e-13
#> Without.knots:Smooth:Quadratic:ARIMA
#> 6.714e-09
#> With.knots:Smooth:Quadratic:ARIMA
#> -4.587e-09
#> Without.knots:With.knots:Smooth:Quadratic:ARIMA
#> -4.477e-14
#>
#>
#> $`Ensembled based on summed weight`
#>
#> Call:
#> stats::lm(formula = Series ~ Without.knots + With.knots + Smooth +
#> Quadratic + ARIMA)
#>
#> Coefficients:
#> (Intercept) Without.knots With.knots Smooth Quadratic
#> -29.792396 0.001736 0.001500 -0.028442 0.568967
#> ARIMA
#> 0.026470
#>
#>
#> $`Ensembled based on weight of fit`
#> Time Series:
#> Start = 1
#> End = 320
#> Frequency = 1
#> 1 2 3 4 5 6
#> -6.0247547 -6.3547327 -6.5850396 -6.5737599 -6.3937495 -6.0719406
#> 7 8 9 10 11 12
#> -5.5895815 -4.9778759 -4.2256078 -3.3216903 -2.2626748 -0.9914928
#> 13 14 15 16 17 18
#> 0.2895600 1.7573324 3.3650593 5.0807941 6.9488668 8.9362281
#> 19 20 21 22 23 24
#> 11.0460341 13.2829096 16.0443395 18.4417233 21.4578305 23.8269686
#> 25 26 27 28 29 30
#> 27.0609031 30.0420945 32.8042197 36.5400203 40.3415168 42.5791831
#> 31 32 33 34 35 36
#> 46.0881330 48.4376580 53.8127498 55.5023082 59.3360127 61.5881275
#> 37 38 39 40 41 42
#> 64.8439983 67.7135312 71.3609714 76.9728191 78.9493186 82.0051487
#> 43 44 45 46 47 48
#> 86.1054842 89.0943541 93.6293003 97.1082804 101.5772934 106.1365207
#> 49 50 51 52 53 54
#> 110.5507480 115.6308235 121.8530071 129.8862730 132.4073799 142.7352298
#> 55 56 57 58 59 60
#> 146.7659476 152.4659445 159.4141459 162.7414360 168.8383440 172.2976848
#> 61 62 63 64 65 66
#> 185.2701095 190.7645989 195.9501374 205.0570004 209.8462030 213.7087874
#> 67 68 69 70 71 72
#> 224.9758931 222.7884865 227.0501694 241.3052728 243.0281387 238.0774623
#> 73 74 75 76 77 78
#> 244.2877583 248.4088547 245.6192692 250.2675407 248.6768835 250.8448545
#> 79 80 81 82 83 84
#> 243.8355423 256.6326888 256.3070679 262.1549102 273.5461842 278.0054637
#> 85 86 87 88 89 90
#> 273.3990680 274.7727132 279.5736400 277.0167408 285.2527105 296.8907957
#> 91 92 93 94 95 96
#> 284.7536531 298.9318337 312.7328903 301.2019791 312.3313682 306.1379805
#> 97 98 99 100 101 102
#> 313.8512584 319.1455996 316.0897304 320.2609506 310.7044016 315.9616801
#> 103 104 105 106 107 108
#> 345.8210957 336.9605017 361.3619896 366.3733156 358.1919090 360.1132067
#> 109 110 111 112 113 114
#> 372.8536536 369.3310323 374.4515214 384.4909855 376.9885431 380.2535544
#> 115 116 117 118 119 120
#> 371.9983424 390.8752649 380.5832617 388.1550686 382.0786255 377.9219521
#> 121 122 123 124 125 126
#> 385.8557198 369.2527964 379.3628489 382.9756292 394.0550508 381.7188804
#> 127 128 129 130 131 132
#> 361.8135977 367.1683817 360.9805054 365.7039705 365.2867906 358.1223432
#> 133 134 135 136 137 138
#> 356.1250318 356.5771351 361.7181848 358.1114446 362.4079328 356.9212443
#> 139 140 141 142 143 144
#> 368.6487079 366.9748419 363.8119295 365.4759529 355.1057061 355.1482724
#> 145 146 147 148 149 150
#> 357.5598788 354.4599011 356.4698881 351.0469725 334.8933358 337.8477904
#> 151 152 153 154 155 156
#> 341.3964601 337.9129952 323.3542535 323.4852975 317.1477099 306.1769046
#> 157 158 159 160 161 162
#> 297.5181290 289.2069419 283.6632076 289.5853295 281.8858889 286.8405546
#> 163 164 165 166 167 168
#> 289.1981741 289.0512618 282.9669366 295.8413034 287.9122388 283.0957152
#> 169 170 171 172 173 174
#> 279.6243748 274.4678108 272.3160852 279.3878626 278.5758144 288.4262142
#> 175 176 177 178 179 180
#> 280.9560506 269.7999513 288.0227038 269.0391366 265.0571913 257.0820382
#> 181 182 183 184 185 186
#> 241.0887979 238.4653349 221.9358139 222.0292887 212.9338583 209.5370230
#> 187 188 189 190 191 192
#> 216.1138215 211.6211149 203.4624211 203.0873974 201.0826704 196.1977446
#> 193 194 195 196 197 198
#> 200.3010542 208.6830635 199.4481603 199.9525784 200.0467430 197.3538213
#> 199 200 201 202 203 204
#> 194.1080697 196.4791555 190.9145569 188.7986445 187.1617150 191.4236869
#> 205 206 207 208 209 210
#> 191.8128486 188.3469672 197.6327172 197.4050839 192.2726909 192.4156385
#> 211 212 213 214 215 216
#> 195.1466918 189.0211044 188.9989639 191.2052809 194.3735371 196.6284188
#> 217 218 219 220 221 222
#> 193.6157396 191.2696905 192.9693199 186.7300149 191.3753458 192.1847815
#> 223 224 225 226 227 228
#> 193.0753338 190.6062363 193.6352204 193.4655916 199.0439841 202.8724939
#> 229 230 231 232 233 234
#> 207.9211939 206.4552018 214.0735054 210.8325881 205.1738577 209.0216255
#> 235 236 237 238 239 240
#> 207.3792009 203.0882060 198.8098472 201.9832870 201.0126013 196.8550214
#> 241 242 243 244 245 246
#> 202.0692982 209.6050696 212.8520345 218.6331034 220.9176561 223.6140111
#> 247 248 249 250 251 252
#> 227.0933322 228.7066333 231.2212973 248.0434349 243.2445783 245.7241004
#> 253 254 255 256 257 258
#> 252.1394672 242.4931395 263.4792229 256.3956284 260.8542637 267.2718852
#> 259 260 261 262 263 264
#> 267.1462103 267.3755753 267.4524343 273.9471302 278.1939147 281.6268790
#> 265 266 267 268 269 270
#> 290.2306409 286.4788699 288.6241460 300.6076293 299.6634525 298.8772705
#> 271 272 273 274 275 276
#> 310.7779364 315.1620336 317.0903774 328.8250357 326.8990487 333.3775822
#> 277 278 279 280 281 282
#> 349.1417965 367.9286199 369.1960822 393.7387321 406.1605208 418.9062788
#> 283 284 285 286 287 288
#> 441.2801957 464.0361049 492.4889178 502.8214664 559.1080591 683.1901238
#> 289 290 291 292 293 294
#> 572.3563430 580.3781304 595.4751261 619.0238001 646.4327985 641.0838898
#> 295 296 297 298 299 300
#> 596.3372144 590.2356383 586.1422694 611.1238270 687.6566932 702.1625305
#> 301 302 303 304 305 306
#> 689.8949983 651.6162286 655.1177773 681.4584995 686.6809527 737.1815820
#> 307 308 309 310 311 312
#> 760.1373443 750.2260868 757.1475599 737.7657492 787.1148702 846.4771779
#> 313 314 315 316 317 318
#> 888.4497761 926.2921077 925.5671412 932.1192235 957.6904907 949.9283267
#> 319 320
#> 988.4351256 1002.5929666
#>
#> $`Unconstrained Forecast`
#> DDf91 Case
#> Linear Linear 277722
#> Semilog Semilog 179540
#> Growth Growth 2721
#> Without Knots Without knots 1606956
#> Smooth spline Smooth Spline 2082046
#> With Knots With knots 991316
#> Polynomial Quadratic Polynomial 380508
#> Lower ARIMA Lower ARIMA -119303
#> Upper ARIMA Upper ARIMA 1074901
#> Ensembled with equal weight Essembled with equal weight 1403498
#> Ensembled based on weight Essembled based on weight 154152
#> Ensembled based on summed weight Essembled based on summed weight 211499
#> Ensembled based on weight of fit Essembled based on weight of fit 1099890
#>
#> $RMSE
#> DDf91 RMSE_f91
#> Linear "Linear" "307.33"
#> Semilog "Semilog" "310.78"
#> Growrh "Growth" "475.6"
#> Without knots "Without knots" "221.72"
#> Smooth Spline "Smooth Spline" "185.4"
#> With knots "With knots" "153.91"
#> Polynomial "Quadratic Polynomial" "305.21"
#> Lower ARIMA "Lower ARIMA" "169.16"
#> Upper ARIMA "Upper ARIMA" "169.16"
#> Ensembled with equal weight "Essembled with equal weight" "178.28"
#> Ensembled based on weight "Essembled based on weight" "358.35"
#> Ensembled based on weight "Essembled based on summed weight" "357.6"
#> Ensembled based on weight of fit "Essembled based on weight of fit" "193.29"
#>
#> $`Unconstrained forecast Plot`
#>
#> $Date
#> [1] "Jan 14, 21 - Nov 29, 21"
#>
#> $`Constrained Forecast`
#> Model
#> Linear Linear
#> Semilog Semilog
#> Growth Growth
#> Smooth spline 80% Without knots 80%
#> Smooth spline 95% Without knots 95%
#> Without knots 80% Smooth Spline 80%
#> Without knots 95% Smooth Spline 95%
#> With knots 80% With knots 80%
#> With knots 95% With knots 95%
#> Polynomial 80% Quadratic Polynomial 80%
#> Polynomial 95% Quadratic Polynomial 95%
#> ARIMA 80% ARIMA 80%
#> ARIMA 95% ARIMA 95%
#> Essembled with equal weight 80% Essembled with equal weight 80%
#> Essembled with equal weight 95% Essembled with equal weight 95%
#> Essembled based on weight 80% Essembled based on weight 80%
#> Essembled based on weight 95% Essembled based on weight 95%
#> Essembled based on summed weight 80% Essembled based on summed weight 80%
#> Essembled based on summed weight 95% Essembled based on summed weight 95%
#> Essembled based on weight of fit 80% Essembled based on weight of fit 80%
#> Essembled based on weight of fit 95% Essembled based on weight of fit 95%
#> Confirmed cases
#> Linear 277722
#> Semilog 179540
#> Growth 2721
#> Smooth spline 80% 4723
#> Smooth spline 95% 875749
#> Without knots 80% 15471
#> Without knots 95% 871333
#> With knots 80% 25306
#> With knots 95% 861881
#> Polynomial 80% 406348
#> Polynomial 95% 430240
#> ARIMA 80% 2356
#> ARIMA 95% 876588
#> Essembled with equal weight 80% 17358
#> Essembled with equal weight 95% 867253
#> Essembled based on weight 80% 75091
#> Essembled based on weight 95% 168687
#> Essembled based on summed weight 80% 220605
#> Essembled based on summed weight 95% 357114
#> Essembled based on weight of fit 80% 8815
#> Essembled based on weight of fit 95% 868036
#>
#> $`Constrained forecast Plot`
#>
#> $`Fitted plot`
#>
#> $`Estimated coefficients`
#>
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | Linear without knots | Linear with knots | ARIMA | Quadratic polynomial | Linear | Semilog | Growth |
#> +=======================================+======================+===================+==================+======================+===========+=============+==========+
#> | (Intercept) | -396.533*** | 190.135** | | 147.912** | 66.474+ | -393.533*** | 3.323*** |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | (49.313) | (65.159) | | (51.749) | (34.549) | (88.025) | (0.155) |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | splines = bs(Series, knots = NULL)1 | 2212.594*** | | | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | (142.576) | | | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | splines = bs(Series, knots = NULL)2 | -912.054*** | | | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | (90.701) | | | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | splines = bs(Series, knots = NULL)3 | 1621.249*** | | | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | (77.788) | | | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | splines = bs(Series, knots = BREAKS)1 | | -634.663*** | | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | | (99.062) | | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | splines = bs(Series, knots = BREAKS)2 | | 1741.962*** | | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | | (90.134) | | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | splines = bs(Series, knots = BREAKS)3 | | -1477.319*** | | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | | (110.314) | | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | splines = bs(Series, knots = BREAKS)4 | | 1199.904*** | | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | | (74.529) | | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | ar1 | | | 0.620*** | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | | | (0.112) | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | ar2 | | | -0.642*** | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | | | (0.083) | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | ar3 | | | -0.272*** | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | | | (0.075) | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | ar4 | | | -0.134+ | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | | | (0.068) | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | ar5 | | | -0.170** | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | | | (0.065) | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | ma1 | | | -1.141*** | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | | | (0.096) | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | ma2 | | | 0.791*** | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | | | (0.100) | | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | Series | | | | 0.150 | 1.668*** | | 0.011*** |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | | | | (0.744) | (0.187) | | (0.001) |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | I(Series^2) | | | | 0.005* | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | | | | (0.002) | | | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | log(Series) | | | | | | 152.232*** | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | | | | | | | (18.050) | |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | Num.Obs. | 320 | 320 | 319 | 320 | 320 | 320 | 320 |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | R2 | 0.584 | 0.709 | | 0.212 | 0.201 | 0.183 | 0.342 |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | R2 Adj. | 0.580 | 0.705 | | 0.207 | 0.198 | 0.180 | 0.340 |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | AIC | 4375.0 | 4262.5 | 4197.1 | 4577.6 | 4580.0 | 4587.1 | 1121.6 |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | BIC | 4393.9 | 4285.2 | 4227.3 | 4592.6 | 4591.3 | 4598.4 | 1132.9 |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | Log.Lik. | -2182.518 | -2125.272 | | -2284.776 | -2286.996 | -2290.570 | -557.783 |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | F | 147.900 | 192.015 | | 42.606 | 79.919 | 71.130 | 164.982 |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | RMSE | 221.72 | 185.40 | 169.16 | 305.21 | 307.33 | 310.78 | 1.38 |
#> +---------------------------------------+----------------------+-------------------+------------------+----------------------+-----------+-------------+----------+
#> | x | | | 0.75932820959595 | | | | |
#> +=======================================+======================+===================+==================+======================+===========+=============+==========+
#> | + p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001 |
#> +=======================================+======================+===================+==================+======================+===========+=============+==========+
#>